High-Strength Steels for Plastic Design · Table 1, are (TULT, the ultimate stress and the percent elongation at failure. ... ASTM-A36 36 0.0012 0.014 900 65 25 ASTM cA441 50 0.0017

)

II,I

High-StrengthSteels forPlastic Design

PETER F. ADAMS

REPRINTED FROM

AS ENGINEERING JOURNALPublished by American Insti te 0 Steel Construction, 101 Park Ave., New York, N. Y. 10017

High-Strength Steels for Plastic DesignPETER F. ADAMS

This paper was presented at the AISC National Engineering Conference, Boston, Mass., in April, 7966.

ESH ' 700 ksi

, 900 ksi

Table 1

Figure 7

0.03E

values for other pertinent properties. The modulus ofelasticity, E, for both materials has been taken as 29,500ksi. 6 The strain at the onset of strain-hardening is€SH" The properties not shown in Fig. 1, but listed inTable 1, are (TULT, the ultimate stress and the percentelongation at failure.

It is apparent that any differences in behavior be­tween members of the two types of steel must be creditedto the changes in yield stress and strain-hardeningmodulus.

It has been shown that the variation in propertiesexhibited by the steels in the structural group consideredwill not influence the in-plane behavior of these steels.7

In other words, members constructed of any of thesesteels will successfully redistribute bending momentsafter the formation of plastic hinges and will fail by the

(Ty€y €SH ESH

(Tuil. %ksi ksi Elongotion

ASTM-A36 36 0.0012 0.014 900 65 25

ASTM cA441 50 0.0017 0.021 700 75 23

Peter F. Adams is Associate Professor of Civil Engineering at theUniversity of Alberta, Edmonton, Alberta, Canada.

CURRENT DESIGN SPECIFICATIONS permit the use ofplastic design procedures for low carbon structural steelsonly.l This restriction· is justified because the researchon which the specifications were based was performedmainly on steels of this family.2 .

Since 1962 a project has been in existence at LehighUniversity which has as its aim the extension of plasticdesign procedures to include the high-strength steelshaving yield stresses up to 50 ksi. This group of steelsincludes the following ASTM designations: A440,A441 and A242.

The object of this paper is to compare the behaviorof members of high-strength steel (in this case ASTM­A441) with that of low-carbon structural steel members(A36 or A7) and to draw conclusions concerning thesuitability of the high-strength steels for plastic design.Since the low carbon steels, henceforth called structuralsteels, have been used for many years in plasticallydesigned structures, it will be assumed that properlydesigned members of these steels will behave in asatisfactory manner.3 The high-strength steel memberswill th'en be evaluated using the behavior of the structuralsteel members as a reference.

MATERIAL PROPERTIES

The initial portions of the stress-strain «(T-€) curves ofthe A36 and A441 steels are shown in Fig. 1. Thesecurves are plotted from average values taken from tensile

. tests performed in connection with various projects.4• 5

The most obvious difference in the two curves is theincreased yield stress, (Ty, shown by the A441 specimen.The length of the inelastic plateau for both specimens isapproximately 12€y, where €y represents the yieldstrain (€y = (TylE). The other factor of importance isthe lower strain-hardening modulus, ESH' for the A441material. Table 1 summarizes the properties of thecurves shown in Fig. 1, and in addition includes typical

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AISC ENGINEERING JOURNAL

10.0

14.0

14 'IF 78A441 .

(b)

+

9.710.5

14.0

Figure 2

Figure 3

(a)

8 'IF 31A 36

12.0

dimensionalized as PIPv where Pv is the load whichwould cause the member to yield (Pv = A(Tv)' Thestrain has been nondimensionalized as flfv. The firstspecimen is of A7 steel and has a bit ratio of 17.18 Thusit is considered acceptable under the plastic designspecifications. Local buckling was not observed untilthe average strain was 13 f v. The second specimen wasof A441 material and it also had a bit ratio of 17.4

For this material the bit ratio was above the limit givenby equation (1) and therefore the section could not be

(1)

formation of a mechanism. For both the structuraland high-strength steels termination of the in-planerotation will be marked by local and lateral buckling.8

In the case of beam-columns this may be preceeded byunloading of the member due to excessive bending. 9

CROSS-SECTIONAL PROPERTIES

The residual stress distribution in a member causedby rolling and subsequent cooling has been carefullydocumented. A typical distribution is shown in Fig.2alo for a section of A36 steel. The average compressiveresidual stress at the flange tips is 12.4 ksi or 0.34 (Tv.The distribution for a section of A441 steel is shownin Fig. 2b. In this case the average compressive residualstress at the flange tips is 11.6 ksi or 0.23 (Tv. It has beennoted in other investigations that the magnitude of thecompressive residual stress appears to be independentof the yield stress level. The residual stress distributionsshown in Fig. 2 are due to the rolling process; recentinvestigations indicate that the continuous cold straight­ening process used for the smaller sections, may alterthe distribution shown in Fig. 2 and may, in fact, alterthe measured material properties. For the present how­ever, the distributions shown may be considered astypical.

One other factor that may be considered as a prop­erty of the cross-section is the local buckling strengthof the member. For a section to be admissible undercurrently used design specifications it is required thatthe flange plate be capable of sustaining strains up tostrain-hardening before local buckling occurs.1 For A36steel the flange width-to-thickness ratio, bit, must beless than 17 for the section to be admissible; for A441the corresponding limitation is 14.11 In general terms,the expression for the limiting value is

b

where GSH is the torsional rigidity of the material inthe strain-hardening range. GSH is directly proportionalto E SH and thus the local buckling limit depends notonly on (Tv but also on the strain-hardening modulus.

The importance of the plate geometry is illustratedclearly by the results of a concentric stub column test.A typical specimen is shown ready for testing in Fig. 3.In this test the specimen is subjected to a compressivestrain which is applied uniformly over the cross-sections.The specimen is loaded until the material has yielded.Further deformation occurs with little change in loaduntil local buckling of the plate elements is observed.At this stage the specimen is unable to maintain theload and gradual unloading proceeds with additionaldeformation. Figure 4 shows the load-strain P-f curvesfor two specimens. In Fig. 4 the load has been non-

151

OCTOBER/1966

used in a plastically designed structure. Local bucklingwas observed at an average strain of 4.0 f ll for thisspecimen. The conclusion of the test is shown in Fig. 5.The distortion of the cross-section is evident in thisphotograph.

For most cases, the development of severe localbuckling marks the initiation of unloading. Sinceportions of members used in plastically designed struc­tures must be strained into the strain-hardening range-

IP IR IR IP(0) [I===±I===±I===1 I

(c)~ PL'M

Figure 6

to allow redistribution of the bending moment-the plate elements of such members must bestocky. This is to ensure that local buckling will notoccur before the strain in the material reaches thestrain-hardening. In the cross-sections which follow,it will be assumed that the selected members meet therequirements of equation (1). Similar requirements mustmeet web geometry.2

o 4 B

Figure 4

Figure 5

12

152

MEMBER BEHAVIOR

A member designed according to plastic theory mustdeliver the full plastic moment, MrJ> and maintain thismoment while the hinge rotates a sufficient amount toensure the development of a mechanism. The abilityto deform inelastically is a function of the adequacy ofthe bracing system together with the material properties.

BEAMS UNDER UNIFORM MOMENT

A beam under uniform moment has been the subjectof much research performed in order to study the be­havior of members in the inelastic range. 3 • 4. 13 Thisloading condition was selected because in the elasticrange it represents one of the more critical loadingconditions and is perhaps deceptively simple to analyze.

A typical test arrangement is shown in Fig. 6a.The beam is suspended at its third points by high tensilesteel rods which resist the reactions, R, and is loaded ateach end with a load, P, applied by means of hydraulicjacks. These jacks are operated off a common pressurecircuit to ensure equal loads. The specimen is bracedlaterally at the ends and third points by knife-edgebraces which allow deformations in the plane of theapplied loads but prevent any out-of-plane movement.This is shown in Fig. 6b. The bending moment distri­bution is shown in Fig. 6c. The central span, under theuniform moment, is critical with respect to lateralbuckling. The adjacent spans act as elastic restraints.A photograph of a test in progess is shown in Fig. 7.

The beam is deformed until the moment in thecentral critical span reaches M p • Under further in-planedeformation, the yielding process decreases the stiffnessof the critical span thus magnifying any initial lateral

AISC ENGINEERING JOURNAL

L

(2)

1612B

. Figure 9

4a

1.0

tMMp

0.5 A36

A441 0

K~ ... I 0.56E,,1 +--E SH

In equation (2), K is the effective length and isassumed equal to 0.54 if the adjacent spans are elastic,and 0.8 if the adjacent spans are fully yielded. Forbeams braced at spacings less than that given by equation(2), unloading will be triggered by local buckling, muchin the manner as beams under moment gradient. Thiswill be discussed in the following section.

ference in behavior is due to the change in yield strengthand the difference in the strain-hardening moduli ofthe two materials.

The inelastic rotation which a given section IS

capable of delivering can be related to the materialproperties and the unbraced length. The optimumcapacity is obtained by spacing the lateral bracing sothat the compression flange of the member will bucklelocally. For this type of buckling, the results occur onlywhen the applied in-plane strains are at the same rotationthat will produce large lateral deflections and subsequentlocal buckling as described above.s This will be termedthe optimum bracing spacing and is given by:

BEAMS UNDER MOMENT GRADIENT

The behavior of a beam under moment gradient willbe discussed with respect to the simply supported beamshown in Fig. lOa. The bending moment distribution isshown in Fig. lOb, and the corresponding curvaturedistribution in Fig. lOco

For a beam loaded in this manner, the maximummoment under the load point may be greater than Mf)due to the influence of strain-hardening. As the load isincreased, the member deforms inelastically, causingthe yielded zone to spread on either side of the loadpoint. The extent of the yielded zone at a particularstage in loading is shown by the heavy line in Fig. lOb.Figure 8

Figure 7

deflections present in the compression flange. Thisaction continues until the large lateral deflections, pro­duced by the in-plane deformations, lead to local buck­ling of the compression flange. This is usually followedby a drop off in moment capacity.s The local buckledarea of a typical test specimen is shown in Fig. 8.

In Fig. 9 the moment rotation (M-8) curves aregiven for two specimens, one of A7 material and thesecond of A441 stee1.4 , 13 Both specimens were 10VF25shapes with an unbraced length of 35rv' where rvis the weak axis radius of gyration of the section. InFig. 9 the moment is nondimensionalized as M/M f)and the rotation as 8/8f), where 8f) is the rotation at theend of the central span. It is assumed that the materialis ideally elastic (8f) = M f)L/2EI). Local buckling oc­curred in the A36 member at a rotation of 14.4 8f) and forthe A441 member at a rotation of 5.6 8f)' Again the dif-

153

OCTOBER 11966

,-------------------------------------------------- --

The curvatures within this yielded zone, and thus thestrains, are greatly increased over the elastic values andthe distribution may be approximated by that shown inFig. 10c.

For a section just meeting the requirements ofequation (1) the member would deform inelasticallyuntil the yielded length reaches a value sufficient toallow a local buckle to form. The formation of the localbuckles is shown in Fig. 10d. At this point the responseof the member would fall below that predicted by in­plane considerations and the member would unloadgradually. A photograph of a specimen showing theformation of a local buckle under the load point isgiven in Fig. 11. The moment-rotation curve is shownin Fig. 12. The increase in moment above M p due tothe strain-hardening capacity of the material has not

MMp

Figure 72

It should be noted that although test results are notavailable for structural carbon steel members which arecomparable with the behavior of the A441 member,the test results are compared with a theory that hasbeen checked against the behavior of structural carbonsteel members in other similar situations, and arefound to be satisfactory. This will also be the case forthe test results which follow.

been considered in the theoretical prediction shown bythe dashed line.

The behavior characterized by the response curveof Fig. 12 is probably the best that can be obtained andis independent of the bracing spacing. If the unbracedlength is excessive, the member will buckle laterallybefore the formation of a local buckle. To prevent this,it is recommended that the unbraced length be limitedto:

:Ii I 11

~

I

~Figure 70

L 0.77r

\.l;;; (3)

Figure 77

AISC ENGINEERING JOURNAL

154

BEAM-COLUMNS

The early research on plastic design concentrated onlow, rigid frame structures in which the axial forceswere relatively smalJ.2 The beam under its variousloading and restraint conditions was the primary researchtopic. In a multi-story frame, however, the behaviorof the beam-columns is of great importance and inrecent years experimental and theoretical investigationshave treated this subject. 9, 14, 15

For convenience, the beam-columns considered willbe subjected to a constant axial force and an appliedend-moment. The latter is the variable load parameter.Unloading of the member occurs when the appliedend-moment drops off with increasing deformation.

Three representative loading conditions, to whichthe beam-column may be subjected, are shown in Fig.

----------- - ----

0.040.030.02

8

Figure 15

0.01o

OCTOBER/1966

SUBASSEMBLAGES FOR BRACED FRAMES

Although the behavior of individual members has beenwell documented, it is only recently that investigationshave been oriented toward the behavior of large framesloaded into the inelastic range.J7 Design methods forsuch frames have been established, however, using as abasis the behavior of subassemblages or groups ofmembers which are characteristic of the structure con-

for the beam under moment gradient may be applied.For most columns used in multi-story frames this be­havior is typical. For more slender or highly loadedmembers the end-moment may not reach M pc .

The symmetrical, single curvature case shown inFig. 13 represents the critical loading condition for abeam-column. In this case the secondary moments,produced by the axial load acting through the in-planedeflection of the member, will reduce the bendingcapacity of the section, 0 that the maximum momentwill be less than M pc. Fortunately this severe case isuncommon in multi-story frames, but may arise wherecheckerboard loading is considered.

Recent experiments on high-strength beam-columnshave dealt with this loading case. 16 Figure 14 shows thetest setup. The beam-column i erected in the testingmachine and the axial load is applied through the headof the machine. The end-moments, which deform themember in a symmetrical single curvature mode, areapplied by means of an independent hydraulic jackacting through stub beams, which deliver the jack forceseccentrically to the specimen.

Figure 15 presents the M -(} curve of a high-strengthbeam-column. The response, experimentally deter­mined, is predicted adequately by a theory based on thematerial properties. This is typical of the structuralcarbon steels and accounts for the increased strengthof the A441 material. In order to obtain the optimumresponse of the member, the bracing provisions rec­ommended for beams subjected to uniform momentwere used.

155

0.2

Mrh

MMPc ~

ilp0.1

Ip

Figure 13

Figure 14

8

13. The response of the member under the variousconditions, as characterized by its M-(} curve, is alsoshown in this figure.

For loading conditions other than that producingsymmetrical single curvature, the response of the mem­ber is very similar to that of a beam. As the member isdeformed the end-moment increases until it reaches M pc •

At this point further deformation may occur at a con­stant moment. Unloading occurs only when the yieldedlength (at the point of maximum moment) reaches alength sufficient for the formation of a local buckle.To attain this optimum response the bracing rules used

8

= 8Joint

I ~umnsM~

: + 8• Beams

Figure 20

M

sidered.18 For example, consider the braced frame shownin Fig. 16. The frame is subjected to a uniformly dis­tributed live load on alternate bays and stories so thatthe columns are deformed into a symmetrical singlecurvature mode.

A characteristic subassemblage for this frame isshown in Fig. 17. It consists of the column AB restrainedby the beam and column segments which frame into thetop and bottom joints. The column is deformed by theunbalanced moments due to the checkerboard loadingon the beams acting together with the axial load from thestories above.

For simplicity in testing, the subassemblage ismodified to that shown in Fig. 18. The head of thetesting machine is used to apply the axial load P and theend-moments are applied by a hydraulic jack whichproduces a force, F. This is delivered to the beam­columns th~ough stub beams, producing the deformedshape shown by the dashed lines.

A photograph of a typical subassemblage under testis shown in Fig. 19. In this photograph the three sets oflateral braces on the column can be seen, as well as theknife edge braces used to hold the restraining beamsagainst lateral movement. The tower used to providereaction points for the restraining beams is seen in theright side of the photograph. This tower is bolted to thelaboratory floor. The hydraulic jack, which acts throughthe stub beams is shown on the left side of the figure.This is connected in series with a dynamometer used tomeasure the jack force.

The applied moment M j is resisted partly by thecolumn, with a resisting moment Me> and partly by therestraining beam. The condition at a braced joint isshown in Fig. 20a. The resisting moment of the beam isdenoted as M B and the joint rotation as O. To maintainequilibrium, the applied moment must be balanced by

Figure 78

Heod ofTestin9Pf": Mochine

~Floor of Testin9Machine

------,A ...\I1DeflecledI ShopeIIIB E-- --- .....

Figure 79

Figure 76

DB

Figure 77

c

E

156

AISC ENGINEERING JOURNAL

I I--,.------ ----- f---+t

~ R12 B16.5 (A36)

F!I

10'-0" II

6 W'"25II

(A441)

ItR tFIi

--'----------- --+

300~'Bo .02 .04

.04 e.02--

+ 1000

2'-6"

Figure 21

7'-6"

the sum of the resIsting moments for a given Jomtrotation, O. Compatibility at the joint reg uires that thebeam and column-end rotations be the same as that of thejoint. Thus the response of the structure may be taken asthe sum of the responses of the column and the restrain­ing beam as shown in Fig. 20b.

In the design method proposed for braced multi­story frames, it is desirable that the beam reach itsmaximum moment at a smaller rotation than thecolumn.19 Thus the unloading of the joint would beprecipitated by the unloading of the column.

With this in mind, the subassemblage shown mFig. 21 was tested. The column was a 6V'F25 section ofA441 steel while the restraining beams were 12816.5sections of A36 steel. The dimensions of the subas­semblage were chosen for correlation with a full scalebraced frame test. 17 The axial load on the column washeld at 0.6 Pu'

The results of the subassemblage test are shown inFig. 22. This figure shows the M-O response of the beam,the column and finally the total joint response. Theresponses of the subassemblage components werepredicted by methods developed to predict the be­havior of structural carbon steel structures, modified toaccount for the increased strength of the A441 beam­column. The agreement is excellent.

Figure 23 is a photograph which shows the sub­assemblage after test. Unloading of the subassemblagewas precipitated by the unloading of the column due toexcessive bending. The restraining beams continued toaccept end-moment even after local buckling had

Figure 22

Figure 23

157

OCTOBER/1966

800

Figure 26

400

Me (kip-in)

o

SUMMARY AND CONCLUSIONS

The main points of this paper are summarized asfollows:

1. Differences in behavior between members of high­strength steel and members of carbon structural steel

were chosen so that the frame would exhibit significantsway effects.

The theoretical prediction for the behavior of theframe is shown by the dashed lines in Fig. 25. Thisprediction accounts for 1. the secondary momentsproduced by the sway of the frame, 2. the reduction instiffness due to the formation of plastic hinges, and 3.the increase in strength due to strain-hardening afterthe formation of a mechanism.

The experimental results are shown by the datapoints connected by the solid lines in Fig. 25. Theagreement between theory and experiment is excellent.The frame failed in a combined mechanism, forming twohinges in the leeward column, one at the base of thewindward column and one in the beam at the wind­ward load point. The localized behavior at one of thehinge locations in the high-strength column is shown inthe form of a load-moment (H-M) curve in Fig. 26.Again the agreement between experiment and theory isexceedingly good.

Figure 27 shows the frame after testing. It was inexcellent condition even' after considerable inelasticdeformation. The column hin'ge area is shown in Fig. 28,The extent of yielding is considerable, but due to stockyplate geometry, local buckling did not occur.

6

Figure 25

flI I I I Hn -7-I // /

3W W W 3W

~ ~ . ~160

~l S1~H

lor 25.4 (A36)

r-~tH12.0 Me

8'-9 5W 18.5 (A441)

H(kips)

8.0

2~2' 15'

Figure 24

4.0

H

SUBASSEMBLAGES FOR UNBRACED FRAMES

The suba:s~emblage used as a model for the design ofunbr<;tced multi-story frames has not yet been tested asa unit. 20 However,' A441 beam-columns in sway situa­tions have been tested and their performance has beensatisfactory.21

The test frame is shown in Fig. 24. The frame isa fixed-base portal· frame with concentrated verticalloaqs over the column tops and at the quarter points ofthe beam. These loads are held constant, during the test,at a value which produces an axial load in the columnsof 0.26 Py • The sway deformation is produced by a hori­zontal load, H, applied at the mid-depth ~f the beam.The test frame simulates, in a general \yay, the situationin anyone story of a multi~story frame subjected' tohorizontal load due to wind or earthquake.

The columns are SW18.S sections of A441 steelwhile the beam is a 10125.4 of A36 steel. The dimensions

occurred. There was no sign of local or lateral bucklingof the beam-column.

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AISC ENGINEERING JOURNAL

------------------------

are caused by the increased yield point and the reducedstrain-hardening modulus of the high-strength steel.If these factors are taken into account, the inelasticbehavior of high-strength steel members may be pre­dicted. This investigation was restricted to structuralsteels having yield stress levels up to 50 ksi and yielding inthe same manner as the structural low carbon steels.

2. The magnitude of the residual stresses due to therolling process appears to be independent of the yieldstress of the material. Solutions based on a maximumcompressive residual stress of 0.3 O'v (which has beenaccepted for the A36 steels) would in general be con­servative when applied to steels having higher yieldstresses.

3. In designing for the higher yield stress level andreduced strain-hardening modulus, flange plates ofmembers, used in plastically designed structures, shouldmeet the requirements of equation (1). Correspondinglimitations are being developed for web plates.

4. Lateral bracing for members under uniform momentand for those under moment gradient may be spacedaccording to equations (2) and (3).

5. Beams under uniform moment and momentgradient have been tested and the results of these testshave adequately predicted theory. (The members wereof A441 steel.)

6. Beam-columns of A441 steel have been tested aloneand as segments of subassemblages under both non­sway and sway conditions. In all cases the results havebeen predicted successfully by a theory which is basedon the same principles as those used in the design of low­carbon steel members.

7. It appears certain that, in the near future, high­strength steels will be used in plastically designed

Figure 27

159

structures. The problems that remain to be solved be­fore this advance can be taken, are those associated withweb buckling, connection behavior and overall framestability. Work is in progress in these areas.

ACKNOWLEDGMENT

The research reviewed in this paper is being conductedat Fritz Engineering Laboratory, Department of CivilEngineering, Lehigh University. Sponsorship for theprogram is provided by the American Institute of SteelConstruction, the American Iron and Steel Institute, theBureau of Ships, the Bureau of Yards and Docks and theWelding Research Council. This research project isgratefully acknowledged, as are the major contributionsof the past and present, by the staff at Fritz Laboratory.

REFERENCES

1. Manual of Steel Construction, American Institute of SteelConstruction 6th ed., 1963.

2. ASCE and WRC, Commentary on Plastic Design inSteel, ASCE Manual No. 41, 1961.

3. Lay, M. G. The Experimental Bases for Plastic Design,Welding Research Council Bulletin No. 99, Sept., 1964.

4. Adams, P. F., Lay, M. G., and Galambos, T. V. Experi­ments on High Strength Steel Members, Welding Re­search Council Bulletin No. 110, Nov., 1965.

Figure 28

OCTOBER/1966

14. Galambos, T. V., and Ketter, R. L. Columns under Com­bined Bending and Thrust, Trans. ASCE, Vol. 126, p. 1,1961.

15. Ojalvo, M. Restrained Columns, Proc. ASCE, Vol. 86,EM5, Oct., 1960.

16. Aglietti, R. A., Lay, M. G., and Galambos, T. V. Testson A36 and A441 Steel Beam-Columns, Fritz LaboratoryReport No. 278.14, June, 1964.

17. Yura, J. A., and Driscoll, G. C., Jr. Plastic Design ofMulti-story Buildings-A Progress Report, AISC Engi­nerring Journal, Vol. 2, No.3, July 1965.

18. Levi, V., Driscoll, G. C., Jr., and Lu, L. W. StructuralSubassemblages Prevented from Sway, Proc. ASCE, Vol.91, ST5, Dec., 1965.

19. Lu, L. W. Examples of Braced Frame Design, Chapter 72,Lecture Notes on Plastic Design of Multi-story Frames,Fritz Laboratory Report No. 273.20, Summer, 1965.

20. Daniels, J. H., and Lu, L. W. The Subassemblage Methodof Designing Unbraced Multi-story Frames, Fritz Labo­ratory Report, No. 273.37, March, 7966.

21. Arnold, Peter, Adams P. F., and Lu, L. W. Behavior of aHybrid Portal Frame, Fritz Laboratory Report No. 273.78.(In Preparation.)

Compression and TensionProc. ASTM, Vol. 41, pp.

5. Thurlimann, Bruno New Aspects Concerning InelasticInstability of Steel Structures, Proc. ASCE, Vol. 86, ST1,Jan., 1960.

6. Johnston, B., and Opila, F.Tests of Structural Alloys,552-570, 1941.

7. Adams, P. F. Plastic Design in High Strength Steel,Ph.D. Dissertation, Lehigh University. (In Preparation.)

8. Lay, M. G., and Galambos, T. V. Inelastic Steel BeamsUniform Moment, Proc. ASCE, Vol. 91, ST6, Dec., 1965.

9. Van Kuren, R. C., and Galambos, T. V. Beam ColumnExperiments, Proc. ASCE, Vol. 90, ST2, Apr., 1964.Beedle, L. S., and Tall, L. Basic Column Strength, Proc.ASCE, Vol. 86, ST7, July, 1960.Galambos, T. V. Beams, Chapter 3, Lecture Notes onPlastic Design of Multi-story Frames, Fritz LaboratoryReport No. 273.20, Summer 1965.HaaiJer, G., and Thurliman, B. Local Buckling of WideFlange Shapes, Fritz Laboratory Report No. 205E5, Dec.,1954.Lee, G. C., and Galambos, T. V. Post Buckling Strengthof Wide Flange Beams, Proc. ASCE, Vol. 88 EM1, Feb.,1962.

10.

11.

12.

13.

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